S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 1 Maxwell’s equations are: D = ρ ∇· B =0 ∇· ∂B E + =0 ∇ ∧ ∂t ∂D H = J ∇ ∧ − ∂t In these equations: ρ and J are the density and flux of free charge; E and B are the fields that exert forces on charges and currents (f is force per unit volume): f = ρE + J B; or for a point charge F = q(E + v B); ∧ ∧ D and H are related to E and B but include contributions from charges and currents bound within atoms: D = ǫ E + P H = B/µ M; 0 0 − P is the polarization and M the magnetization of the matter. −7 −2 µ0 has a defined value of 4π 10 kgmC . S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 2 Maxwell’s equations in this form apply to spatial averages (over regions of atomic size) of the fundamental charges, currents and fields. This averaging generates a division of the charges and currents into two classes: the free charges, represented by ρ and J, and charges and cur- rents in atoms, whose averaged effects are represented by P and M. We can see what these effects are by substituting for D and H: E =(ρ P) /ǫ ∇· −∇· 0 B =0 ∇· ∂B E + =0 ∇ ∧ ∂t ∂E ∂P B ǫ µ = µ J + + M . ∇ ∧ − 0 0 ∂t 0 ∂t ∇ ∧ We see that the divergence of P generates a charge density: ρb = P −∇ ·∂P and the curl of M and temporal changes in P generate current: J = + M. b ∂t ∇ ∧ Any physical model of the atomic charges and currents will produce these spatially averaged effects (see problems). CWPP 22/2/2010 S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 3 This suggests that if we do not make the spatial average we can treat all charges and currents on the same basis, and obtain the fundamental classical equations relating charges, currents and electromagnetic fields, the Microscopic Maxwell Equations: E = ρ/ǫ M1 ∇· 0 B =0 M2 ∇· ∂B E + =0 M3 ∇ ∧ ∂t ∂E B ǫ µ = µ J M4 ∇ ∧ − 0 0 ∂t 0 These are the equations we shall be working with from now on, and we shall make no further reference to D, H, P or M. The physical interpretation is provided by the force density: f = ρE + J B. ∧ We next look at three simple consequences. S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 4 Local Conservation of Charge Take 1/µ div M4+ǫ ∂/∂t M1 J + ∂ρ/∂t = 0. 0 0 →∇· This equation is known as the continuity equation and implies local conservation of charge. Consider an arbitrary volume V in a current flow. The total charge inside V at time t is Q(t) = ρ(r,t) d3r ZV If charge is locally conserved then the only way in which charge can leave V is by flowing through the bounding surface S: dQ ∂ρ = d3r = J dS = J d3r. − dt − ∂t · ∇· ZV ZS ZV This must hold for an arbitrary volume, so the integrand must vanish: ∂ρ + J =0 ∂t ∇· This ensures the local conservation of charge, and is the prototype for other local conserva- tion laws, which have a density term, a flux term, and in general a loss or source term as well if the quantity is not absolutely conserved but can transform into something else. CWPP 22/2/2010 S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 5 Conservation of Energy (Poynting’s Theorem) We can find a local conservation law for energy in the electromagnetic field. Take B (M3) and subtract E (M4): · · ∂B ∂E B ( E) E ( B) + B + ǫ µ E + µ E J =0 · ∇ ∧ − · ∇ ∧ · ∂t 0 0 · ∂t 0 · We can combine the terms in the first bracket as (E B). ∇· ∧ ∂E 1 ∂E2 We can rewrite the partial derivatives using E = and similarly for the B terms. · ∂t 2 ∂t Thus we have a conservation law with a loss/source term: ∂u S + + E J =0 ∇· ∂t · where: 1 S = E B is Poynting’s vector and represents the energy flux in the field; µ0 ∧ ǫ 1 u = 0 E2 + B2 is the energy density in the field; 2 2µ0 E J represents the rate per unit volume of energy loss from the EM field to the matter, · represented by the current (or from matter to field if negative). S8: Covariant Electromagnetism MAXWELL’S EQUATIONS 6 Wave Equations for E and B Eliminate B by taking ∂/∂t of M4 and subtracting the curl of M3: ∂2E ∂J ǫ µ E = µ . − 0 0 ∂t2 −∇∧∇∧ 0 ∂t Using the vector identity curl curl = grad div 2, and substituting from M1: − ∇ ∂2E ∂J 2E ǫ µ = µ + ρ/ǫ ∇ − 0 0 ∂t2 0 ∂t ∇ 0 which is the inhomogeneous wave equation with wave speed c where 1 c2 = ǫ0µ0 ∂J and the source term is µ + ρ/ǫ . 0 ∂t ∇ 0 ∂2B Similarly we can eliminate E: 2B ǫ µ = µ J. ∇ − 0 0 ∂t2 − 0∇ ∧ CWPP 22/2/2010 S8: Covariant Electromagnetism COVARIANCE 7 The word COVARIANT, in this context, means the same as form invariant: some transformation of the quantities in the equations produces no change in the form of the equations. Some of the transformations are induced by physical operations (rotations, changes of veloc- ity) that can be understood in either an active or passive sense; for example the system is rotated in space (ACTIVE) or the same system is described relative to two different co-ordinate systems. (PASSIVE) I shall take ALL TRANSFORMATIONS IN A PASSIVE SENSE. We shall discuss the covariance of Maxwell’s equations under the following transformations: Change of Units (!) Rotation (including Reflection) Duality Transformation Gauge Transformation Lorentz Transformation S8: Covariant Electromagnetism UNITS 8 In Maxwell’s equations there are four places we can put constants. However, to understand all the historical arguments we must assume the existence of magnetic charge and current, which increases the number of constants to six: E = αρ ∇· B = κρ ∇· m ∂B E + β = λJ ∇ ∧ ∂t − m ∂E B γ = δJ ∇ ∧ − ∂t The six constants α, β, γ, δ, κ and λ are not all independently variable. If we re-work the derivation of the wave equations we find the coefficient of the time derivia- tive is βγ: 1 βγ = c2 This is an absolute requirement. CWPP 22/2/2010 S8: Covariant Electromagnetism UNITS 9 If we re-work the conservation of electric charge we find ∂ρ αγ + δ J =0 ∂t ∇· so if the unit of charge in J is the same as in ρ then δ = αγ λ = κβ. and similarly for magnetic charge Most systems of units assume that E (B) is the force on a unit electric (magnetic) charge. However, if we re-work Poynting’s theorem we find γ ∂E2 β ∂B2 (E B) + + + δE J + λB J =0 ∇· ∧ 2 ∂t 2 ∂t · · m so we require the coefficients of E J and B J to be the same: · · m δ = λ. This leaves two degrees of freedom: The value of α The way that 1/c2 is factored between β and γ. Various considerations have seemed relevant at different times. S8: Covariant Electromagnetism UNITS 10 Rationalized and Unrationalized Units If we use M1 to derive Coulomb’s law we find αq q F = 1 2 ˆr 4πr2 so we have a choice as to whether the 4π apears in the equation, or in the solution. Gm m Maxwell, following the example of Newton’s law of gravitation F = 1 2 chose to put a r2 factor of 4π in α. (But note he did not use an analogue of G — see below.) It was later recognized (Heaviside) that the factor of 4π appears in solutions with spherical symmetry, and a factor of 2π in problems with cylindrical symmetry (like the B field of a straight wire carrying a current), so it seemed more ‘rational’ to put the factors of 4π in the solution. On the whole this idea has found favour, and the modern fashion is for ‘rationalized’ rather than the older ‘unrationalized’ units. CWPP 22/2/2010 S8: Covariant Electromagnetism UNITS 11 Dimensions In mechanics it is normally assumed (Gauss) that there are three physical dimensions, and thus base units, of mass, length and time. Units for all other quantities, such as velocity or force, are derived from these. In any physical equation the dimensions must be equal in all terms. How to fit Electromagnetism into this scheme has caused the most intense debate. In fact the number of dimensions used in describing a physical situation is a matter of choice, convenience and personal preference. We can choose to increase the number of dimensions: for example an atmospheric model may have different units for height and distance, but that will introduce a new physical constant with dimensions height/distance. We can reduce the number of dimensions, and eliminate physical constants: for example temperature is a measure of energy — there is a conversion constant k =1.380 6503(24) 10−23 J K−1. × We can view the speed of light (m s−1), Planck’s constant (JHz−1) or the Avogadro constant (mol−1) in the same way. We could even choose to eliminate mass as a dimension by setting G = 1. S8: Covariant Electromagnetism UNITS 12 Maxwell’s Choices Maxwell used unrationalized units, and in his development of the theory it seemed natural to choose β = 1, and hence γ =1/c2. He also took the view that the number of physical dimensions was naturally three: In all dynamical sciences it is possible to define . units in terms of the three fundamental units of Length, Time and Mass.